Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}2x+6y &= 4 \\ 3x+3y &= 2\end{align*}$
Solution: We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $2$ $\begin{align*}-2x-6y &= -4\\ 6x+6y &= 4\end{align*}$ Add the top and bottom equations. $4x = 0$ Divide both sides by $4$ and reduce as necessary. $x = 0$ Substitute $0$ for $x$ in the top equation. $2( 0)+6y = 4$ $6y = 4$ $6y = 4$ $y = \dfrac{2}{3}$ The solution is $\enspace x = 0, \enspace y = \dfrac{2}{3}$.